8th Grade - Gateway 2

The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding. Students access concepts from a number of perspectives and independently demonstrate conceptual understanding throughout the grade.

The conceptual understanding required in 8.F.A is addressed in Unit 5 as students define, evaluate, and compare functions. For example:

• Lesson 1 begins to develop the “idea of a function as a rule that assigns to each allowable input exactly one output.” This understanding is cultivated in the first Activity of Lesson 1 as student play "Guess My Rule" in an applet. Students enter any input value into Column A of the spreadsheet found in the applet, and the resulting output appears in Column B. Student pairs generate a rule after completing as many iterations as needed. Functions' rules include additive, multiplicative, and exponential patterns.

• In Lesson 2, function language is introduced after students identify examples and nonexamples of functions in order to extend the idea that the output is dependent on the input. In the first Activity, students are prompted: “A number is 5. Do you know its square?” and the anticipated response is given as, “Yes, the square of 5 is 25.” Another prompt asks, “The square of a number is 16. Do you know the number?” resulting in the anticipated response, “No, there are two different numbers whose square is 16, namely 4 and -4.” The term “function” is not introduced until the second Activity in which students use function language to express whether the given scenarios from the previous Activity are functions. The corresponding exemplar statements from the given examples are, “Yes, the square of a number depends on the number,” and “No, knowing the square of a number does not determine the number.”

• In Lessons 3 through 7, students use and compare verbal descriptions, tabular and graphic representations of functions, as well as equations. In the second Activity in Lesson 4, students are given three unlabeled continuous graphs in order to make connections between representations. They choose the matching equation and context (8.F.3), use the context to identify the dependent and independent variables, and finally, use the graph to identify the output when the input is 1 and interpret what that tells you about each situation in prompt 3 (8.F.1). The graphs include a non-linear representation and two linear functions, one with a positive slope and one with a negative slope.

Cluster 8.G.A addresses congruence and similarity using physical models, transparencies, or geometry software and is found in both Units 1 and 2. Unit 1 begins with transformations.

• In Lessons 1 through 6, students spend most of the instructional time either physically moving shapes or imitating that movement on a GeoGebra program. In Lesson 1, students first look at transformations as a way of moving objects in a plane. They define these movements in their own words in Lesson 2 and establish the actual definition of the movements in later lessons.

• Lessons 11 through 13 explore congruence. In the Lesson 12 Warm-Up, students are given a variety of congruent triangles in different orientations and the following prompt: “All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle ABC under a translation.” Students develop the concept that a two-dimensional figure is congruent to another two-dimensional figure if the second can be obtained from the first by a sequence of transformations. The idea of “rotations and reflections usually (but not always) change the orientation of a figure” is discussed here and further explored when students name a sequence of transformations to prove some of the non-shaded triangles congruent to triangle ABC.

• In Lessons 14 through 16, students establish informal arguments about angles. The second Activity in Lesson 14 states, "Lines ℓ and k are parallel, and t is a transversal. Point M is the midpoint of segment PQ.” Students use tracing paper to “Find a rigid transformation showing that angles MPA and MQB are congruent.”

Cluster 8.G.6 builds understanding of the Pythagorean Theorem and is found in Unit 8.

• Lesson 6 Activity 1 prompts students to find the length of the sides of several triangles using a grid, some of which are right triangles and some are not. Through the investigation, students arrive at an understanding of the Pythagorean Theorem and that it only applies to right triangles. After the investigation the Pythagorean Theorem is officially introduced to the students.

The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that they attend to those standards that set an expectation of procedural skill. Materials attend to the Grade 8 expected procedural skills, particularly those related to solving linear equations and systems of equations.

Procedural skills and fluencies are intentionally built on conceptual understanding and the work students have accomplished with operations and equations from prior grades. Opportunities to formally practice developed procedures are found throughout practice problem sets that follow the units, including opportunities to use and practice emerging fluencies in the context of solving problems. Units 1-3 include embedded review of rational number operations, and Units 5-9 include continued practice of content that is developed in Unit 4. According to the Design Principles within the Grade 8 Course Guide within each unit, “Students are systematically introduced to representations, contexts, language, and notation. As their learning progresses, they see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency.” Number Talks included in many Warm-Ups often revisit fluencies developed in earlier grades and specifically relate to the Activities found in the lessons. Additionally, students demonstrate procedural skill and fluency throughout the year in a variety of practice problems. Examples of lesson practice problems follow:

• 8.EE.7 is found in Unit 5.
  • In Lesson 3, students solve and check the solutions to multi-step equations such as 4z+5=-3z−8.
  • In Lesson 7, students solve and check the solution to the multi-step equation -(-2x+1)=9−14x.
  • In Lesson 9 Problem 3, students solve and explain the reasoning leading to the solutions of multi-step equations such as 4(2a+2)=8(2−3a).
• 8.EE.8b is found in Unit 5.
  • In Lesson 1, students solve a system of two equations, both in slope-intercept form, using substitution.
  • In Lesson 3, students graph a system of equations with no solution and then write the equation of each graphed line.
  • In Lesson 4, students solve a system of two equations, both in slope-intercept form.
  • In Lesson 12, students solve a system of two equations, both in slope-intercept form.
  • In Lesson 20, students write and solve a system of equations representing the context given about a bicycle shop inventory in order to find the number of bicycles in the store.

8.EE.7 is found in Unit 4 where students use computational skills to solve linear equations in one variable.

• Lesson 1 through 4 connect to past grades with lessons and activities that incorporate the idea of balancing equations.

• In the Lesson 3 Warm-Up, students match hanger diagrams to equations and variables to their respective shapes within the diagram. Next, students begin to match the first “moves” in solving equations in the first Activity. In the additional activities in Lesson 3, students “[think] about strategically solving equations by paying attention to their structure” when they are presented two student work samples to evaluate and provide recommendations for solving.

• In Lesson 4, there is a mix of tasks that focuses on practicing solving equations such as matching, choosing solution steps, evaluating the work of sample student solution paths, as well as assessing similarities and looking for mistakes.

• In Lesson 5, students move toward a general method for solving linear equations using mental math to solve one-step equations for a variable on one side and then work with a partner to justify their steps with one another between each step.

• In Lesson 6 Strategic Solving, “In this lesson, students learn to stop and think ahead strategically before plunging into a solution method. After a Warm-Up in which they construct their own equation to solve a problem, they look at equations with different structures and decide whether the solution will be positive, negative, or zero, without solving the equation. They judge which equations are likely to be easy to solve and which are likely to be difficult.” In the following lessons students look at situations when an equation has many or no solutions. Once students are introduced to a general procedure for solving equations continuous practice is provided.

Systems of equations (8.EE.8b) are formally introduced in the latter part of Unit 4 after students solve linear equations in one variable (8.EE.7), including writing, solving, and graphing equations as well as deciding what it means for an equation to be true. Students learn to interpret and solve systems of equations in Lessons 12 through 15 in preparation for applying their procedural knowledge in Lesson 16.

• In the Lesson 12 Warm-Up, students take a situation and features of the graphs without actually graphing, to develop fluency with the vocabulary and begin to visualize where systems of equations may have solutions. All tasks involve graphing equations and discussing solutions in context of graphs. The term “system of equations” is introduced.

• In the Lesson 14 Warm-Up, students use substitution strategies to mentally solve systems of equations. In Activity 1, students analyze the structure of a system of equations before deciding on an efficient solution path. The equations lend themselves to suggesting substitution as the first step toward finding a solution and develop the procedure for substituting an expression in place of a variable.

• In Lesson 15, students write and interpret systems of equations from contexts with rational coefficients and continue to practice solving using various methods, including: “[Solving] the systems to find the number of solutions.; [Using] the slope and y-intercept to determine the number of solutions.; [Manipulating] the equations into another form, then compare the equations. [Noticing] that the left side of the second equation in system C is double the left side of the first equation, but the right side is not.”

The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, both routine and non-routine, presented in a context in which the mathematics is applied.

Work with applications of mathematics occurs throughout the materials in ways that enhance the focus on major work and when standards call for application in real-world or mathematical contexts. The Grade 8 Course Guide states: “Students have opportunities to make connections to real-world contexts throughout the materials. Carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end.” Connections between clusters and application extensions are also found in the multiple-day lessons found in optional Unit 9.

Cluster 8.F.B addresses students using functions to model relationships between quantities and is found in Unit 5. Lessons 5 through 11 are specifically identified as addressing 8.F.B and use a variety of applications as students model relationships using functions. Students identify and create tables, graphs, and equations modeling relationships.

• The Lesson 6 Warm-Up states: “The purpose of this Warm-Up is for students to realize there are different dependent variables that can be used when making a model of a context, and the choice of which we use affects how a graph of a function looks.” Students view five photographs of a dog taken at equal intervals of time and two graphs representing the scenario. Both graphs have the same independent variable but look dramatically different. Students determine how the dependent variable represents the perspective of the graph.

• In the Lesson 10 Warm-Up, students are asked to share what they notice on a graph of temperature data during different parts of the day. In Activity 1, they use piecewise linear graphs to find information about the real-life situation they represent. In Activity 2, students analyze a situation to calculate the rate of change.

• In Lesson 11 Activity 1, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data. Students use an applet to reason about the height of the water in a given cylinder. They graph the relationship and then explain the meaning of specific points in their recorded data. Students need to apply the context of the given cylinder to the graph to determine, “What would the endpoint of the graph be?” In the next phase of the Activity, students compare this relationship to ones in which the radius of the cylinder has been modified and explain how the slope is less steep in the given graphic representation.

Standard 8.EE.8c addresses students solving real-world and mathematical problems leading to two linear equations in two variables and is found in Unit 4. Lesson 16 includes opportunities for students to investigate applications of systems of equations.

• In Activity 1, students solve problems involving real-world contexts. For the first problem, students find the time at which two friends will meet if they are cycling toward one another. In the second problem, they determine how many grapefruits are sold if students are selling both grapefruits and nuts. The price of both items is given as well as the total number of items sold and the total money made in the fundraiser. In the third problem, students find the number of hours Andre and Jada must work to make the same amount of money when working different jobs and getting paid different rates. In each problem, students “explain or show [their] reasoning.” After engaging in these problems, students create their own situation and solve. These scenarios are then exchanged for other pairs of students to solve.

The instructional materials for Open Up Resources 6-8 Math, Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

There is evidence that the curriculum addresses standards, when called for, with specific and separate aspects of rigor and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized.

Examples of conceptual understanding include:

• A conceptual understanding of transformations is developed in Unit 1. Students begin this unit by simply describing how one figure moves to another. Students use tracing paper or GeoGebra to make “the moves” and answer the questions with these visual, hands-on tools. In Lesson 3 Activity 1, students are given four figures on a grid and told: “In Figure 1, translate triangle ABC so that A goes to A′. In Figure 2, translate triangle ABC so that C goes to C′. In Figure 3, rotate triangle ABC 90o counterclockwise using center O. In Figure 4, reflect triangle ABC using line l.” Activities similar to this are done repeatedly throughout the beginning lessons of Unit 1. Eventually, students use this information to draw conclusions about congruent figures, angles, and similar figures (8.G.A).

• In Unit 5 Lesson 6, students demonstrate conceptual understanding of graphs as they relate to context and determine the scale to represent the independent and dependent variables. Students create graphs for a given story/context determining the scale. Discussion questions include: “Which quantity is a function of which? Explain your reasoning;" "Based on your graph, is his friend’s house or the park closer to Noah's home? Explain how you know;" and, "Read the story and all your responses again. Does everything make sense? If not, make changes to your work.”

Examples of procedural skill include:

• In Unit 5 Lesson 5, students develop procedural skill solving equations with one variable. During the Warm-Up, students solve the following equations mentally: 5−x=8, -1=x−2, -3x=9, and -10=-5x. In the first Activity, students are given a card with a more complex equation on it and told to work with a partner, taking turns after each step to solve the equation. They solve four cards in all.

• In the Unit 5 Lesson 16 Warm-Up, students apply computational skills using the formula for volume to solve: “27=(1⁄3)h, 27=(1⁄3)r², 12π=(1⁄3)πa, 12π=(1⁄3)πb².” In the first Activity, students practice finding relevant information and completing tables to apply the volume formula to find the value of unknown dimensions.

Examples of application include:

• In Unit 5 Lesson 21, students practice 8.G.9 as they solve a variety of mathematical problems involving finding the volume cones, cylinders, rectangular prisms, and spheres in the given figures.

• In Unit 2 Lesson 13 Activity 3, students apply their understanding of both proportional relationships (7.RP.2) from the previous grade and arguments establishing facts about similar triangles and angle relationships (8.G.5) to estimate the height of a tall object that cannot be measured directly. Students use their learning from the unit and the previous activities to devise a method to estimate, justify, and test.

All three aspects of rigor are balanced throughout the course, including the unit assessments. There are multiple lessons where two or all three of the aspects are connected. For example:

• In Unit 5 Lessons 3 through 7, students develop their understanding of functions by comparing multiple representations. The majority of Activities use real-world contexts with frequent opportunities for students to interpret functions and their representations in specific contexts.

• The Practice Problems available for each lesson are strategically arranged so that procedural practice problems lead to opportunities for students to practice and develop proficiency and skills for a concept and engage with more complex application-level problems. On average there are six problems included in the practice problems. Procedural practice, visual representations, contexts, and/or standard methods of solving said problems are present.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All eight MPs are clearly identified throughout the materials, with few or no exceptions. The Math Practices are initially identified in the Teacher Guide under the narrative descriptions of each unit within the Course Information. For example:

• Unit 2 Dilations, Similarity and Introducing Slope excerpts include: “Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students use and extend their knowledge of geometry and geometric measurement.” Further, in the narrative it states, “They use the definition of “similar” and properties of similar figures to justify claims of similarity or non-similarity and to reason about similar figures (MP3).

• In Unit 6 Associations in Data, the narrative states: “[Students] return to the data on height and arm span gathered at the beginning of the unit, describe the association between the two, and fit a line to the data (MP4). The third section focuses on using two-way tables to analyze categorical data (MP4). Students use a two-way frequency table to create a relative frequency table to examine the percentages represented in each intersection of categories to look for any associations between the categories. Students also examine and create bar and segmented bar graphs to visualize any associations.”

Within a lesson, the MPs are identified within the teacher narratives accompanying the lesson in general or before each of the activities. Lesson narratives often highlight when a Math Practice is particularly important for a concept or when a task may exemplify the identified Practice. For example:

• MP8: The Unit 1 Lesson 14 Lesson introduction states, “One Mathematical Practice that is particularly relevant for this lesson is MP8. Students will notice as they calculate angles that they are repeatedly using vertical and adjacent angles and that often they have a choice which method to apply. They will also notice that the angles made by parallel lines cut by a transversal are the same and this observation is the key structure in this lesson.”

• MP5: The Unit 2 Lesson 2 introduction discusses developing the idea of dilations by providing tools for students: “As with previous geometry lessons, students should have access to geometry toolkits so they can make strategic choices about which tools to use (MP5).”

• MP3: The Unit 4 Lesson 5 narrative accompanying Activity 1 states, “The goal of this Activity is for students to build fluency solving equations with variables on each side. Students describe each step in their solution process to a partner and justify how each of their changes maintains the equality of the two expressions (MP3).”

• MP8: The Unit 5 Lesson 3 Warm-Up narrative states: “The purpose of this Warm-Up is for students to use repeated reasoning to write an algebraic expression to represent a rule of a function (MP8).”

The MPs are used to enrich the mathematical content and are not treated separately from the content in stand-alone lessons. MPs are used to enrich the mathematical content and are discussed within narratives as pertaining to the learning target or specific task at hand. The narratives are used to support deepening a teacher’s understanding of the standard itself as the teacher is provided direction regarding how the content is connected to the MP. For example:

• MP6: In the Unit 1 Lesson 13 Overview, the connection of MP6 to 8.G.2 is explained: “One of the mathematical practices that takes center stage in this lesson is MP6. For congruent figures built out of several different parts (for example, a collection of circles), the distances between all pairs of points must be the same. It is not enough that the constituent parts (circles for example) be congruent: they must also be in the same configuration, the same distance apart. This follows from the definition of congruence: rigid motions do not change distances between points, so if Figure 1 is congruent to Figure 2 then the distance between any pair of points in Figure 1 is equal to the distance between the corresponding pair of points in Figure 2.”

• MP7: In Unit 9 Lesson 2, students extend their knowledge of transformations into the creation of a tessellation. The introduction to the second Activity states, “Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.”

The MPs are not identified in the student materials; however, they are highlighted in the Teacher Guide in the narrative provided with each Activity. For example, in the student task accompanying Unit 9 Lesson 2 Activity 2, the student facing directions state (see previous bullet for teacher facing information): “With your partner, choose one of the six shapes in the toolbar that you will both use. 1) Select the shape tool by clicking on it. Create copies of your shape by clicking in the work space. 2) When you have enough to work with, click on the Move tool (the arrow) to drag or turn them. 3) If you have trouble aligning the shapes, right click to turn on the grid.” After they have finished, the discussion prompts lead students to find structure in the patterns, “Compare your tessellation to your partner’s. How are they similar? How are they different?”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the unit and lesson narratives, as appropriate, when they relate to the overall work. They are also explained within individual activities, when necessary. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP1 Make sense of problems and persevere in solving them.

• In the Unit 1 Lesson 6 first Activity, students “make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need.” The following instructions are provided for the teacher: “Tell students they will continue to describe transformations using coordinates. Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of two. Provide access to graph paper. In each group, distribute a problem card to one student and a data card to the other student. They need to know which transformations were applied (i.e., translation, rotation, or reflection). They need to determine the order in which the transformations were applied. They need to remember what information is needed to describe a translation, rotation, or reflection.”

• In Unit 5 Lesson 21 Activity 3, students engage in another Info Gap activity. One student is given a card that has a question related to volume equations of cylinders, cones, and spheres. The other student is given a card with all the information needed to answer the question. The student with the question asks the student with the data a series of questions that will give them the necessary information to solve the problem. It may take several rounds of discussion if their first requests do not yield the information they need, creating a situation in which students have to persevere to solve a problem.

MP2 Reason abstractly and quantitatively.

• In Unit 5 Lesson 7 Activity 4, students compare properties of functions represented in different ways. Students are given a verbal description and a table to compare and decide whose family traveled farther over the same time intervals. The purpose of this activity is for students to continue building their skill interpreting and comparing functions.

• In Unit 7, students reason abstractly and quantitatively to solve problems involving operations with exponents. For example, in the first Activity of Lesson 2, students are given three base ten blocks: a hundred block, a ten block, and a one block. They must answer several questions: “If each small square represents 102, then what does the medium rectangle represent? The large square?” Additional questions change the chosen square and power of 10. The visual element provides both an abstract and quantitative entry point to the problem as students are introduced to the Laws of Exponents.

MP4 Model with mathematics.

• In Unit 2 Lesson 13, students model a real-world context with similar triangles to find the height of an unknown object. Students examine the length of shadows of different objects to find that a proportional relationship exists between the height of the object and the length of its shadow. Students use their knowledge of similar triangles and the hypothesis that the rays of sunlight making the shadows are parallel to justify the proportional relationship between the object and its shadow. Students then go outside and make their own measurements of different objects and the lengths of their shadows and use this technique to estimate the height of these objects.

• In Unit 6 Lesson 8 Activity 2, students model and analyze data related to arm span and height measurements that were gathered in a previous lesson. The students create a scatter plot, identify and explain outliers, and explain whether the equation y = x is a good fit for the data. This activity is an opportunity for students to explore data and practice several strategies for comparing bivariate data.

MP5 Use appropriate tools strategically.

• Throughout Unit 1 lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to select appropriate tools and use them strategically to solve problems. Lessons in this unit ensure the full depth of this MP by emphasizing choice. For example, the Lesson 1 Overview states: “To make strategic choices about when to use which tools (MP5), students need to have opportunities to make those choices. Apps and simulations should supplement rather than replace physical tools.” In the Lesson 14 Warm-Up, the narrative includes the following guidance for teachers: “Some students may wish to use protractors, either to double check work or to investigate the different angle measures. This is an appropriate use of technology (MP5), but ask these students what other methods they could use instead.”

• In Unit 7 Lesson 10, students have opportunities to use digital tools and number lines, using them to think about how to rewrite expressions with exponents. In the second Activity, students are given a table showing how fast light waves or electricity can travel through different materials and a number line applet labeled 0, 1 x 108, 2 x 108, ...9 x 108, 1 x 109 with a magnifier that will expand the space between any two consecutive numbers. The students must convert the given speeds to a usable format and plot them on the given number line(s) as precisely as possible.

MP7 Look for and make use of structure.

• In Unit 7 Lesson 3, students look for patterns when powers of 10 are raised to a power. The students write powers of 10 in expanded form and simplify expressions. The structure of the activity allows students to develop an understanding of powers raised to powers. In Lesson 6, students analyze the structure of exponents to make sense of expressions with multiple bases.

• Prior to Unit 2 Lesson 11 second Activity, students have already generated a rule to determine whether or not a point with coordinates (x,y) lies on a certain line when the line represents a proportional relationship. In this activity, students find a rule to determine if a point (x,y) lies on a line that does not pass through (0,0). Students use the structure of a line and properties of similar triangles to investigate rules relating pairs of coordinates on a line.

MP8 Look for and express regularity in repeated reasoning.

• In Unit 3 Lesson 8, students write equations of lines using y = mx + b. In this lesson and the ones that lead up to this, students develop their understanding based on repeated reasoning about equations of lines. In a previous lesson, students wrote an equation of a line by making generalizations from repeated calculations using their understanding of similar triangles and slope. Additionally, they have written an equation of a linear relationship by reasoning about initial values and rates of change and have graphed the equation as a line in the plane. In this lesson, they develop the idea that any line in the plane can be considered a vertical translation of a line through the origin.

• In the Unit 7 Lesson 1 second Activity, students explore the idea that repeated division by two is equivalent to repeated multiplication by ½. This allows students to make sense of negative exponents later in Lesson 3, where students look for patterns when bases of 10 are raised to a power in the given chart. Finally, in Lesson 5, students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication of 1/10.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and/or analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Students are consistently asked to explain their reasoning and compare their strategies for solving in small group and whole class settings.

• In the Unit 2 Lesson 6 Cool-Down, students analyze and correct a series of transformations and a dilation intended to provide a similar figure. In the Unit 2 Lesson 7 first Activity, students analyze two figures that are claimed to be similar. Students justify or deny the claim; they are building reasoning as to what characteristics are found in polygons that are not similar.

• In the Unit 5 Lesson 4 first Activity, students are asked, “For each function: What is the output when the input is 1? What does this tell you about the situation? Label the corresponding point on the graph. Find two more input-output pairs. What do they tell you about the situation? Label the corresponding points on the graph.” Questions such as these are present throughout the lessons, providing students the opportunity to construct viable arguments in both verbal and written form.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate, explaining their reasoning to each other.

• The Unit 3 Lesson 5 Warm-Up provides guiding questions in the Activity Synthesis to help students practice MP3. This strategy is used repeatedly throughout the teacher materials. “To involve more students in the conversation, consider asking: Who can restate ____’s reasoning in a different way? Did anyone solve the problem the same way but would explain it differently? Did anyone solve the problem in a different way? Does anyone want to add on to _____’s strategy? Do you agree or disagree? Why?”

• The Unit 6 Lesson 10 second Activity provides guidance to the teacher during the observation of small groups using data displays to find a bivariate association. “As students work, identify groups that use the different segmented bar graphs to explain why there is an association between the color of the eraser and flaws…Select previously identified groups to share their explanation for noticing an association.”

• In Unit 3 Lesson 11 Activity 2, students explore vertical and horizontal lines in the coordinate plane. Teachers are prompted to: “...pause their work after question 2 and discuss which equation makes sense and why.”

• In the Unit 1 Lesson 11 Overview of the first Activity, teachers are supported to facilitate a discussion to promote student debate so teachers can identify reasons and construct arguments as well as how they critique/analyze responses from others. The teacher guidance also provides ways to help students analyze/critique other’s arguments if it doesn’t occur naturally by providing the teacher examples of what to say/suggest to promote more discourse: “For each pair of shapes, poll the class. Count how many students decided each pair was the same or not the same. Then for each pair of shapes, select at least one student to defend their reasoning.” Sample responses are provided for the teacher.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 8 meet expectations that the materials attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the teacher materials, the Grade 8 Glossary is located in the Teacher Guide within the Course Information section. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples. In the student materials, the Grade 8 Glossary is accessible by a tab within each unit or in the bottom margin of each lesson page. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples.

• Both the unit and the lesson narratives contain specific guidance for the teacher as to best methods to support students to communicate mathematically. Within the lesson narratives, new terms are in bold print and explained as related to the context of the material.

• Unit 1 develops the concept of Rigid Transformations. In the initial lessons, students use their own words to describe moving one figure to another. As the unit progresses, the students build their understanding of transformations and start naming these movements with the proper terminology (translations, reflections, rotations).

• Unit 6 Lessons 1 through 6 develop the concept of scatter plots and the related language. In Lessons 1 through 3, students are introduced to the definition of scatter plot, and then explore data represented by a scatter plot and the meaning of a specific point on a graph. In Lesson 4, the terms outlier and line of best fit are introduced, and in Lesson 5, students learn to explain trends in scatter plots followed by the slope of the line of best fit in Lesson 6. Each lesson builds on the initial definition of scatter plot until students work with all aspects of them, understand the concept, and use the related language effectively.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.